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A generalization of the Liouville–Arnol'd theorem

Published online by Cambridge University Press:  24 October 2008

G. E. Prince ,
G. B. Byrnes ,
J. Sherring  and
S. E. Godfrey
G. E. Prince
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia
G. B. Byrnes
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia
J. Sherring
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia
S. E. Godfrey
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia
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Abstract

We show that the Liouville-Arnol'd theorem concerning knowledge of involutory first integrals for Hamiltonian systems is available for any system of second order ordinary differential equations. In establishing this result we also provide a new proof of the standard theorem in the setting of non-autonomous, regular Lagrangian mechanics on the evolution space ℝ × TM of a manifold M. Both the original theorem and its generalization rely on a certain bijection between symmetries of the system and its first integrals. We give two examples of the use of the theorem for systems on ℝ2 which are not Euler-Lagrange.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 353 - 370
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Arnol'd., V. I.Mathematical methods of classical mechanics (Springer, Berlin, 1978).Google Scholar
[2]Blair., D. E.Contact manifolds in Riemannian geometry. Lecture Notes in Mathematics 509 (Springer, Berlin, 1976).Google Scholar
[3]Gartan., E.Leçons sur les invariants intégraux (Hermann, Paris, 1922).Google Scholar
[4]Crampin, M. and Pirani., F. A. E.Applicable differential geometry (Cambridge University Press, 1987).CrossRefGoogle Scholar
[5]Crampin, M., Prince, G. E. and Thompson., G.A geometric version of the Helmholtz conditions in time dependent Lagrangian dynamics. J. Phys. A: Math. Gen. 17 (1984), 1437–1447.Google Scholar
[6]Douglas., J.Solution of the inverse problem of the calculus of variations. Trans. Am. Math. Soc. 50, 71–128.CrossRefGoogle Scholar
[7]Eliashberg., Y. Contact 3-manifolds twenty years since J. Martinet's work. Preprint, Stanford University (1992).CrossRefGoogle Scholar
[8]Gluck., H.Global theory of dynamical systems, Lecture Notes in Mathematics 819 (Springer, Berlin, 1980).Google Scholar
[9]Libermann, P. and Marle., C.-M.Symplectic geometry and analytical mechanics (Reidel, Dordrecht, 1987).CrossRefGoogle Scholar
[10]Liouville., J.Journal de Math. 20 (1855), 137.Google Scholar
[11]Sarlet, W., Prince, G. E. and Crampin., M.Adjoint symmetries for time dependent second order equations. J. Phys. A: Math. Gen. 23 (1990), 1335–1347.CrossRefGoogle Scholar
[12]Sheering, J. and Prince., G. E.Geometric aspects of reduction of order. Trans. Am. Math. Soc. 334 (1992), 433–53.Google Scholar
[13]Vaisman., I.Cohomology and differential forms (Marcel Dekker, 1973).Google Scholar